β€’ Necessary for computing summary statistics β€’ Sums of numerical values β€’ Denoted by the capital Greek letter sigma (Ξ£) 𝒏 𝑿 π’Š = π‘ΏπŸ + π‘ΏπŸ + π‘ΏπŸ‘ + β‹― + 𝑿𝒏 π’Š=𝟏 β€’ The summation of the sum (or difference) of two or more terms equals the sum (or difference) of the individual summations β€’ The summation of a constant, c, times a variable, X, equals the constant times the summation of the variable β€’ The summation of a constant, c, from i = 1 to n, equals the product of n and c. β€’ The subscript may be any letter, but the most common are i, j, and k. β€’ The lower limit of the summation may start with any number β€’ The lower limit of the summation is not necessarily a subscript β€’ The sum of the squared values of X is NOT equal to the square of the sum of X β€’ The summation of the square of (X+Y) is NOT equal to the summation of the sum of X2 and the sum of Y2 β€’ The sum of the product of X and Y is NOT equal to the product of the sum of values of X and the sum of values of Y β€’ The sum of the quotient of X and Y is NOT equal to the quotient of the sum of X and the sum of Y β€’ The sum of the square root of X is not equal to the square root of the sum of X β€’ Most common average β€’ Sum of all observed values divided by the number of observations β€’ For ungrouped and grouped data β€’ Population mean (ΞΌ) β€’ Sample mean ( 𝑋 ) β€’ Different formulas for grouped and ungrouped data β€’ Most common measure of central tendency β€’ Uses all observed values β€’ May or may not be an actual value in the data set β€’ May be computed for both grouped and ungrouped data sets β€’ Extreme observations affect the value of the mean β€’ The sum of the deviations of the observed values from the mean is zero. 𝒏 π‘Ώπ’Š βˆ’ 𝑿 π’Š=𝟏 = 𝟎 β€’ The sum of the squared deviations of the observed values from the mean is smallest. β€’ The Weighted Mean β€’ There are weights β€’ Values are not of equal importance β€’ e.g. GWA β€’ The Combined Mean β€’ Mean of several data sets β€’ The Trimmed Mean β€’ Less affected by extreme observations β€’ Order data β€’ Remove a certain percentage of the lower and upper ends β€’ Calculate arithmetic mean β€’ Middle value in an ordered set of observations β€’ Divides ordered set of observations into two equal parts β€’ Positional middle of an array β€’ Grouped and ungrouped data β€’ Extreme values affect the median less than the mean β€’ The median is used when β€’ We want the exact middle value of the distribution β€’ There are extreme observed values β€’ FDT has open-ended intervals β€’ Most frequent observed value in the data set β€’ Small data sets: inspection β€’ Large data sets: array or FDT β€’ Less popular β€’ Gives the most typical value of a set of observations β€’ Few low or high values do not easily affect the mode β€’ May not be unique and may not exist β€’ Several modes can exist β€’ Value of the mode is always in the data set β€’ May be used for both quantitative and qualitative data sets β€’ Data is symmetric and unimodal οƒ  all three measures may be used β€’ Data is asymmetric οƒ  use median or mode (if unique); trimmed mean β€’ Describe shape of data οƒ  use all three Ungrouped 𝟏 𝑿 = 𝒏 Grouped 𝒏 π‘Ώπ’Š π’Š=𝟏 𝑿= π’Œ π’Š=𝟏 π’‡π’Š π‘Ώπ’Š π’Œ π’Š=𝟏 π’‡π’Š Ungrouped β€’ n is odd Grouped 𝑛 𝑀𝑑 = 𝐿𝐢𝐡𝑀𝑑 + 𝐢 𝑀𝑑 = 𝑋 𝑛 +1 2 β€’ n is even 𝑀𝑑 = 𝑋 𝑛 + 𝑋 𝑛 +1 2 2 2 2 βˆ’< 𝐢𝐹𝑀𝑑 βˆ’1 𝑓𝑀𝑑 β€’ Grouped π‘“π‘€π‘œ βˆ’ 𝑓1 π‘€π‘œ = πΏπΆπ΅π‘€π‘œ + 𝐢 2π‘“π‘€π‘œ βˆ’ 𝑓1 βˆ’ 𝑓2 Find the MEAN, MEDIAN, and MODE of the ff: 1. A sample survey in a certain province showed the number of underweight children under five years of age in each barangay: 3 5 6 4 7 8 6 9 10 4 6 7 5 8983455 2. Given the frequency distribution table of scores β€’ Measure of central tendency β€’ Measure of location β€’ Positional middle β€’ Percentiles β€’ Deciles β€’ Quartiles β€’ Divide ordered observations into 100 equal parts β€’ 99 percentiles; roughly 1 percent of observations in each group β€’ Interpretation: P1, the first percentile, is the value below which 1 percent of the ordered values fall. (ETC.) β€’ Ungrouped data β€’ Empirical Distribution Number with Averaging β€’ Weighted Average Estimate β€’ Grouped data β€’ Divide the ordered observations into 10 equal parts β€’ Each part has ten percent of the observations β€’ Divides observations into 4 equal parts β€’ Each part has 25 percent of the observations Q1 D1 P10 D2 P20 Q2 D3 P30 D4 P40 D5 P50 Q3 D6 P60 D7 P70 D8 P80 D9 P90 Empirical β€’ nk/100 is an integer π‘ƒπ‘˜ = 𝑋 π‘›π‘˜ + 𝑋 π‘›π‘˜ +1 100 Weighted Average (𝑛 + 1)π‘˜ =𝑗+𝑔 100 100 2 β€’ nk/100 is not an integer π‘ƒπ‘˜ = 𝑋 π‘›π‘˜ +1 100 π‘ƒπ‘˜ = 1 βˆ’ 𝑔 𝑋 𝑗 + 𝑔𝑋(𝑗 +1) π‘ƒπ‘˜ = πΏπΆπ΅π‘ƒπ‘˜ + 𝐢 π‘›π‘˜ βˆ’< πΆπΉπ‘ƒπ‘˜ βˆ’1 100 π‘“π‘ƒπ‘˜ β€’ The number of incorrect answers on a true-false exam for a random sample of 20 students was recorded as follows: 2 β€’ and 2 find the 90th percentile, 7th decile, and 1st quartile β€’ Given the frequency distribution of scores of 200 students in an entrance exam in college, find the 95th percentile, 4th decile, and 3rd quartile. Scores Freq.